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In
geophysics Geophysics () is a subject of natural science concerned with the physical processes and Physical property, properties of Earth and its surrounding space environment, and the use of quantitative methods for their analysis. Geophysicists conduct i ...
and physical geodesy, a geopotential model is the theoretical analysis of measuring and calculating the effects of
Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
's
gravitational field In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...
(the
geopotential Geopotential (symbol ''W'') is the potential of the Earth's gravity field. It has SI units of square metre per square seconds (m2/s2). For convenience it is often defined as the of the potential energy per unit mass, so that the gravity vect ...
). The Earth is not exactly spherical, mainly because of its rotation around the polar axis that makes its shape slightly oblate. However, a
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics co ...
series expansion In mathematics, a series expansion is a technique that expresses a Function (mathematics), function as an infinite sum, or Series (mathematics), series, of simpler functions. It is a method for calculating a Function (mathematics), function that ...
captures the actual field with increasing fidelity. If Earth's shape were perfectly known together with the exact mass density ρ = ρ(''x'', ''y'', ''z''), it could be integrated numerically (when combined with a reciprocal distance kernel) to find an accurate model for Earth's gravitational field. However, the situation is in fact the opposite: by observing the orbits of spacecraft and the Moon, Earth's gravitational field can be determined quite accurately. The best estimate of Earth's mass is obtained by dividing the product ''GM'' as determined from the analysis of spacecraft orbit with a value for the
gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
''G'', determined to a lower relative accuracy using other physical methods.


Background

From the defining equations () and () it is clear (taking the partial derivatives of the integrand) that outside the body in empty space the following differential equations are valid for the field caused by the body: Functions of the form \phi = R(r)\, \Theta(\theta)\, \Phi(\varphi) where (''r'', θ, φ) are the spherical coordinates which satisfy the partial differential equation () (the Laplace equation) are called spherical harmonic functions. They take the forms: where spherical coordinates (''r'', θ, φ) are used, given here in terms of cartesian (''x, y, z'') for reference: also ''P''0''n'' are the
Legendre polynomials In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
and ''Pmn'' for are the associated Legendre functions. The first spherical harmonics with ''n'' = 0, 1, 2, 3 are presented in the table below. ote that the sign convention differs from the one in the page about the associated Legendre polynomials, here P_2^1(x)=3x\sqrt whereas there P_2^1(x)=-3x\sqrt. :


Formulation

The model for Earth's gravitational potential is a sum where \mu = GM and the coordinates () are relative to the standard geodetic reference system extended into space with origin in the center of the reference ellipsoid and with ''z''-axis in the direction of the polar axis. The zonal terms refer to terms of the form: :\frac \quad n=0,1,2,\dots and the tesseral terms terms refer to terms of the form: :\frac\,, \quad 1 \le m \le n \quad n=1,2,\dots :\frac The zonal and tesseral terms for ''n'' = 1 are left out in (). The coefficients for the n=1 with both m=0 and m=1 term correspond to an arbitrarily oriented dipole term in the multi-pole expansion. Gravity does not physically exhibit any dipole character and so the integral characterizing ''n'' = 1 must be zero. The different coefficients ''Jn'', ''Cnm'', ''Snm'', are then given the values for which the best possible agreement between the computed and the observed spacecraft orbits is obtained. As ''P''0n(''x'') = −''P''0n(−''x'') non-zero coefficients ''Jn'' for odd ''n'' correspond to a lack of symmetry "north–south" relative the equatorial plane for the mass distribution of Earth. Non-zero coefficients ''Cnm'', ''Snm'' correspond to a lack of rotational symmetry around the polar axis for the mass distribution of Earth, i.e. to a "tri-axiality" of Earth. For large values of ''n'' the coefficients above (that are divided by ''r''(''n'' + 1) in ()) take very large values when for example kilometers and seconds are used as units. In the literature it is common to introduce some arbitrary "reference radius" ''R'' close to Earth's radius and to work with the dimensionless coefficients :\begin \tilde &= -\frac, & \tilde &= -\frac, & \tilde &= -\frac \end and to write the potential as


Derivation

The spherical harmonics are derived from the approach of looking for harmonic functions of the form where (''r'', θ, φ) are the spherical coordinates defined by the equations (). By straightforward calculations one gets that for any function ''f'' Introducing the expression () in () one gets that As the term :\frac\frac\left(r^2\frac\right) only depends on the variable r and the sum :\frac\frac\left(\cos\theta \frac\right) + \frac\frac only depends on the variables θ and φ. One gets that φ is harmonic if and only if and for some constant \lambda. From () then follows that :\frac\ \cos\theta\ \frac\left(\cos\theta \frac\right)\ + \lambda\ \cos^2\theta\ +\ \frac\frac\ =\ 0 The first two terms only depend on the variable \theta and the third only on the variable \varphi. From the definition of φ as a spherical coordinate it is clear that Φ(φ) must be periodic with the period 2π and one must therefore have that and for some integer ''m'' as the family of solutions to () then are With the variable substitution :x=\sin \theta equation () takes the form From () follows that in order to have a solution \phi with :R(r) = \frac one must have that :\lambda = n (n + 1) If ''Pn''(''x'') is a solution to the differential equation one therefore has that the potential corresponding to ''m'' = 0 :\phi = \frac\ P_n(\sin\theta) which is rotationally symmetric around the z-axis is a harmonic function If P_^(x) is a solution to the differential equation with ''m'' ≥ 1 one has the potential where ''a'' and ''b'' are arbitrary constants is a harmonic function that depends on φ and therefore is not rotationally symmetric around the z-axis The differential equation () is the Legendre differential equation for which the
Legendre polynomials In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
defined are the solutions. The arbitrary factor 1/(2n''n''!) is selected to make and for odd ''n'' and for even ''n''. The first six Legendre polynomials are: The solutions to differential equation () are the associated Legendre functions One therefore has that : P_n^m (\sin\theta) = \cos^m \theta\ \frac (\sin\theta)


Largest terms

The dominating term (after the term −μ/''r'') in () is the ''J''2 coefficient, the '' second dynamic form factor'' representing the oblateness of Earth: :u = \frac = J_2 \frac \frac (3\sin^2\theta -1) = J_2 \frac \frac (3 z^2 -r^2) Relative the coordinate system illustrated in figure 1 the components of the force caused by the "''J''2 term" are In the rectangular coordinate system (''x, y, z'') with unit vectors (''x̂ ŷ ẑ'') the force components are: The components of the force corresponding to the "''J''3 term" :u = \frac = J_3 \frac \frac \sin\theta \left(5\sin^2\theta - 3\right) = J_3 \frac \frac z \left(5 z^2 - 3 r^2\right) are and The exact numerical values for the coefficients deviate (somewhat) between different Earth models but for the lowest coefficients they all agree almost exactly. For the JGM-3 model ( see below) the values are: : μ = 398600.440 km3⋅s−2 : ''J''2 = 1.75553 × 1010 km5⋅s−2 : ''J''3 = −2.61913 × 1011 km6⋅s−2 For example, at a radius of 6600 km (about 200 km above Earth's surface) ''J''3/(''J''2''r'') is about 0.002; i.e., the correction to the "''J''2 force" from the "''J''3 term" is in the order of 2 permille. The negative value of ''J''3 implies that for a point mass in Earth's equatorial plane the gravitational force is tilted slightly towards the south due to the lack of symmetry for the mass distribution of Earth's "north–south".


Recursive algorithms used for the numerical propagation of spacecraft orbits

Spacecraft orbits are computed by the
numerical integration In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
of the equation of motion. For this the gravitational force, i.e. the
gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
of the potential, must be computed. Efficient recursive algorithms have been designed to compute the gravitational force for any N_z and N_t (the max degree of zonal and tesseral terms) and such algorithms are used in standard orbit propagation software.


Available models

The earliest Earth models in general use by
NASA The National Aeronautics and Space Administration (NASA ) is an independent agencies of the United States government, independent agency of the federal government of the United States, US federal government responsible for the United States ...
and ESRO/ ESA were the "Goddard Earth Models" developed by Goddard Space Flight Center (GSFC) denoted "GEM-1", "GEM-2", "GEM-3", and so on. Later the "Joint Earth Gravity Models" denoted "JGM-1", "JGM-2", "JGM-3" developed by GSFC in cooperation with universities and private companies became available. The newer models generally provided higher order terms than their precursors. The EGM96 uses ''Nz'' = ''Nt'' = 360 resulting in 130317 coefficients. An EGM2008 model is available as well. For a normal Earth satellite requiring an orbit determination/prediction accuracy of a few meters the "JGM-3" truncated to ''Nz'' = ''Nt'' = 36 (1365 coefficients) is usually sufficient. Inaccuracies from the modeling of the air-drag and to a lesser extent the solar radiation pressure will exceed the inaccuracies caused by the gravitation modeling errors. The dimensionless coefficients \tilde = -\frac, \tilde = -\frac, \tilde = -\frac for the first zonal and tesseral terms (using R = and \mu = ) of the JGM-3 model are According to JGM-3 one therefore has that ''J'' = × 6378.1363 × = and ''J'' = × 6378.1363 × = .


See also

* Geoid#Spherical harmonics representation


References


Further reading

* El'Yasberg ''Theory of flight of artificial earth satellites'', Israel program for Scientific Translations (1967) * Lerch, F.J., Wagner, C.A., Smith, D.E., Sandson, M.L., Brownd, J.E., Richardson, J.A.,"Gravitational Field Models for the Earth (GEM1&2)", Report X55372146, Goddard Space Flight Center, Greenbelt/Maryland, 1972 * Lerch, F.J., Wagner, C.A., Putney, M.L., Sandson, M.L., Brownd, J.E., Richardson, J.A., Taylor, W.A., "Gravitational Field Models GEM3 and 4", Report X59272476, Goddard Space Flight Center, Greenbelt/Maryland, 1972 * Lerch, F.J., Wagner, C.A., Richardson, J.A., Brownd, J.E., "Goddard Earth Models (5 and 6)", Report X92174145, Goddard Space Flight Center, Greenbelt/Maryland, 1974 * Lerch, F.J., Wagner, C.A., Klosko, S.M., Belott, R.P., Laubscher, R.E., Raylor, W.A., "Gravity Model Improvement Using Geos3 Altimetry (GEM10A and 10B)", 1978 Spring Annual Meeting of the American Geophysical Union, Miami, 1978 * * * Lerch, F.J., Klosko, S.M., Patel, G.B., "A Refined Gravity Model from Lageos (GEML2)", 'NASA Technical Memorandum 84986, Goddard Space Flight Center, Greenbelt/Maryland, 1983 * Lerch, F.J., Nerem, R.S., Putney, B.H., Felsentreger, T.L., Sanchez, B.V., Klosko, S.M., Patel, G.B., Williamson, R.G., Chinn, D.S., Chan, J.C., Rachlin, K.E., Chandler, N.L., McCarthy, J.J., Marshall, J.A., Luthcke, S.B., Pavlis, D.W., Robbins, J.W., Kapoor, S., Pavlis, E.C., " Geopotential Models of the Earth from Satellite Tracking, Altimeter and Surface Gravity Observations: GEMT3 and GEMT3S", NASA Technical Memorandum 104555, Goddard Space Flight Center, Greenbelt/Maryland, 1992 * * *


External links

* http://cddis.nasa.gov/lw13/docs/papers/sci_lemoine_1m.pdf * http://geodesy.geology.ohio-state.edu/course/refpapers/Tapley_JGR_JGM3_96.pdf {{Webarchive, url=https://web.archive.org/web/20160304084508/http://geodesy.geology.ohio-state.edu/course/refpapers/Tapley_JGR_JGM3_96.pdf , date=2016-03-04 Spaceflight concepts Gravity Earth orbits